Question: What is the greatest common factor of $54z^{3}$, $18z$, and $36z^{2}$ ?
Answer: Let's factor each monomial to its prime factors: $\begin{aligned} 54z^{3}&=(2)(3)(3)(3)(z)(z)(z) \\\\ 18z&=(2)(3)(3)(z) \\\\ 36z^{2}&=(2)(2)(3)(3)(z)(z) \end{aligned}$ We want the largest set of factors that's included in all three monomials. All of the monomials have one factor of $ 2$, two factors of $ 3$, and one factor of $ z$ : $\begin{aligned} 54z^{3}&=( 2)( 3)( 3)(3)( z)(z)(z) \\\\ 18z&=( 2)( 3)( 3)( z) \\\\ 36z^{2}&=( 2)(2)( 3)( 3)( z)(z) \end{aligned}$ This is the greatest common factor: $( 2)( 3)( 3)( z)=18z$